Within Person

1. Single Hop - Within Person

Specify covariance matrix

Need to create a full covariance matrix. Using clubSandwhich package to help Assuming a correlation of 0.85 between timepoints.

Model Selection

Data are now ready for meta-analysis

Need to now decide on how to fit models. Several different structures were trialed in piloting.

Based on piloting best results were achieved with fitting random effects for timepoints, nested within cohorts. Trialled fitting extra random effects e.g. effect size id (es_id), as well as separate models where study group was also a separate random effect, however resulting models were too complex, with indistinguishable random effects and overall a simpler model chosen.

Decision making here revolves around how to fit the timepoint predictor - i.e. what sort of relationship is present between yi and timepoint Different models are generated then using fit statistics (AIC, BIC, AIcc), visual inspection of fit and expected fit based on knowledge to decide on best fit.

5 different shapes of fit tried: - linear - log - polynomial - 3 knot restricted cubic spline - 4 knot restricted cubic spline

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.     AICc.
## 1     Linear 222.3709 -444.7419 -436.7419 -424.5425 -436.4770
## 2        Log 234.8248 -469.6496 -461.6496 -449.4502 -461.3847
## 3   Poly (2) 225.7822 -451.5643 -441.5643 -426.3472 -441.1617
## 4 3 knot RCS 231.0868 -462.1736 -452.1736 -436.9565 -451.7709
## 5 4 knot RCS 258.3506 -516.7012 -504.7012 -486.4795 -504.1298
## 
## [[2]]

Final Model - Single Leg Hop

## 
## Multivariate Meta-Analysis Model (k = 158; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 114)
## inner factor: timepoint_mean (nlvls = 94)
## 
##             estim    sqrt  fixed 
## tau^2      0.0023  0.0477     no 
## rho        0.9715             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 154) = 2846.1799, p-val < .0001
## 
## Number of estimates:   158
## Number of clusters:    114
## Estimates per cluster: 1-5 (mean: 1.39, median: 1)
## 
## Test of Moderators (coefficients 2:4):¹
## F(df1 = 3, df2 = 25.18) = 73.0069, p-val < .0001
## 
## Model Results:
## 
##                                         estimate      se¹      tval¹     df¹ 
## intrcpt                                  -0.3318  0.0177   -18.7525   17.19  
## rcs(timepoint_mean, 4)timepoint_mean      0.0316  0.0023    13.9569    17.2  
## rcs(timepoint_mean, 4)timepoint_mean'    -0.9673  0.0790   -12.2451   21.27  
## rcs(timepoint_mean, 4)timepoint_mean''    1.6875  0.1396    12.0888   21.77  
##                                           pval¹    ci.lb¹    ci.ub¹      
## intrcpt                                 <.0001   -0.3691   -0.2945   *** 
## rcs(timepoint_mean, 4)timepoint_mean    <.0001    0.0268    0.0363   *** 
## rcs(timepoint_mean, 4)timepoint_mean'   <.0001   -1.1314   -0.8031   *** 
## rcs(timepoint_mean, 4)timepoint_mean''  <.0001    1.3978    1.9772   *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
93.8 (92.8 to 94.9) 93.5 (92.3 to 94.7) 94.2 (93.1 to 95.3) NA 106 months
## [1] 35.80413
## [1]  4.099606  8.100000 13.500000 62.400000

2. Triple Hop

Specify covariance matrix

Model Selection

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.     AICc.
## 1     Linear 69.59918 -139.1984 -131.1984 -123.6311 -130.2893
## 2        Log 74.63269 -149.2654 -141.2654 -133.6981 -140.3563
## 3   Poly (2) 71.37856 -142.7571 -132.7571 -123.4011 -131.3285
## 4 3 knot RCS 73.92010 -147.8402 -137.8402 -128.4842 -136.4116
## 5 4 knot RCS 73.95875 -147.9175 -135.9175 -124.8166 -133.8175
## 
## [[2]]

Final Model - Triple Hop

(not crossover)

## 
## Multivariate Meta-Analysis Model (k = 51; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 40)
## inner factor: timepoint_mean (nlvls = 38)
## 
##             estim    sqrt  fixed 
## tau^2      0.0029  0.0536     no 
## rho        0.9425             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 49) = 828.4431, p-val < .0001
## 
## Number of estimates:   51
## Number of clusters:    40
## Estimates per cluster: 1-3 (mean: 1.27, median: 1)
## 
## Test of Moderators (coefficient 2):¹
## F(df1 = 1, df2 = 12.96) = 28.4696, p-val = 0.0001
## 
## Model Results:
## 
##                      estimate      se¹     tval¹     df¹    pval¹    ci.lb¹ 
## intrcpt               -0.1934  0.0267   -7.2552   20.03   <.0001   -0.2490  
## log(timepoint_mean)    0.0509  0.0095    5.3357   12.96   0.0001    0.0303  
##                        ci.ub¹      
## intrcpt              -0.1378   *** 
## log(timepoint_mean)   0.0715   *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
93.5 (92 to 95.1) 96.9 (94.9 to 98.9) 101.5 (98 to 105.2) 28 months 94 months
## [1] 26.33288

3. Triple Crossover Hop

Specify covariance matrix

Model Selection

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.     AICc.
## 1     Linear 39.85827 -79.71653 -71.71653 -65.27286 -70.46653
## 2        Log 42.70753 -85.41506 -77.41506 -70.97139 -76.16506
## 3   Poly (2) 40.39153 -80.78305 -70.78305 -62.86546 -68.78305
## 4 3 knot RCS 42.03468 -84.06936 -74.06936 -66.15177 -72.06936
## 5 4 knot RCS 41.24803 -82.49607 -70.49607 -61.16398 -67.49607
## 
## [[2]]

Final Model - Triple Crossover Hop

## 
## Multivariate Meta-Analysis Model (k = 39; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 29)
## inner factor: timepoint_mean (nlvls = 29)
## 
##             estim    sqrt  fixed 
## tau^2      0.0075  0.0864     no 
## rho        0.9736             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 37) = 1247.9040, p-val < .0001
## 
## Number of estimates:   39
## Number of clusters:    29
## Estimates per cluster: 1-3 (mean: 1.34, median: 1)
## 
## Test of Moderators (coefficient 2):¹
## F(df1 = 1, df2 = 11.29) = 13.3637, p-val = 0.0036
## 
## Model Results:
## 
##                      estimate      se¹     tval¹     df¹    pval¹    ci.lb¹ 
## intrcpt               -0.2404  0.0535   -4.4973   17.01   0.0003   -0.3532  
## log(timepoint_mean)    0.0617  0.0169    3.6556   11.29   0.0036    0.0247  
##                        ci.ub¹      
## intrcpt              -0.1276   *** 
## log(timepoint_mean)   0.0988    ** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
91.7 (88.7 to 94.7) 95.7 (93.1 to 98.4) 101.2 (96.4 to 106.3) 30 months 94 months
## [1] 16.04113

4. Side Hop

Specify covariance matrix

Model Selection

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.     AICc.
## 1     Linear 19.46570 -38.93140 -30.93140 -27.59855 -27.59807
## 2        Log 20.50714 -41.01428 -33.01428 -29.68143 -29.68095
## 3   Poly (2) 19.36695 -38.73390 -28.73390 -24.87095 -22.73390
## 4 3 knot RCS 22.73347 -45.46693 -35.46693 -31.60399 -29.46693
## 5 4 knot RCS 22.51515 -45.03030 -33.03030 -28.78200 -22.53030
## 
## [[2]]

Final Model - Side Hop

## 
## Multivariate Meta-Analysis Model (k = 19; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 13)
## inner factor: timepoint_mean (nlvls = 15)
## 
##             estim    sqrt  fixed 
## tau^2      0.0057  0.0758     no 
## rho        0.9981             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 16) = 264.3766, p-val < .0001
## 
## Number of estimates:   19
## Number of clusters:    13
## Estimates per cluster: 1-3 (mean: 1.46, median: 1)
## 
## Test of Moderators (coefficients 2:3):¹
## F(df1 = 2, df2 = 1.5) = 121.3381, p-val = 0.0219
## 
## Model Results:
## 
##                                        estimate      se¹      tval¹    df¹ 
## intrcpt                                 -0.3477  0.0233   -14.9472   5.86  
## rcs(timepoint_mean, 3)timepoint_mean     0.0248  0.0018    13.7188   2.16  
## rcs(timepoint_mean, 3)timepoint_mean'   -0.0200  0.0020    -9.9507   2.85  
##                                          pval¹    ci.lb¹    ci.ub¹      
## intrcpt                                <.0001   -0.4050   -0.2905   *** 
## rcs(timepoint_mean, 3)timepoint_mean   0.0039    0.0175    0.0321    ** 
## rcs(timepoint_mean, 3)timepoint_mean'  0.0027   -0.0266   -0.0134    ** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
92.8 (87.9 to 98) 95.8 (91.3 to 100.5) 99 (88.1 to 111.3) 15 months 62 months
## [1] -3.899879
## [1]  6.50000 11.08165 18.00000

5. Vertical Hop

Specify covariance matrix

Model Selection

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.     AICc.
## 1     Linear 35.14039 -70.28077 -62.28077 -57.24839 -60.37601
## 2        Log 37.21730 -74.43461 -66.43461 -61.40222 -64.52985
## 3   Poly (2) 35.00802 -70.01605 -60.01605 -53.92167 -56.85815
## 4 3 knot RCS 36.97917 -73.95833 -63.95833 -57.86395 -60.80044
## 5 4 knot RCS 35.15689 -70.31378 -58.31378 -51.24546 -53.37261
## 
## [[2]]

Final Model - Vertical Hop

## 
## Multivariate Meta-Analysis Model (k = 28; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 22)
## inner factor: timepoint_mean (nlvls = 20)
## 
##             estim    sqrt  fixed 
## tau^2      0.0029  0.0541     no 
## rho        0.6086             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 26) = 538.2331, p-val < .0001
## 
## Number of estimates:   28
## Number of clusters:    22
## Estimates per cluster: 1-3 (mean: 1.27, median: 1)
## 
## Test of Moderators (coefficient 2):¹
## F(df1 = 1, df2 = 9.53) = 11.1711, p-val = 0.0080
## 
## Model Results:
## 
##                      estimate      se¹     tval¹     df¹    pval¹    ci.lb¹ 
## intrcpt               -0.2864  0.0496   -5.7796   12.96   <.0001   -0.3935  
## log(timepoint_mean)    0.0594  0.0178    3.3423    9.53   0.0080    0.0195  
##                        ci.ub¹      
## intrcpt              -0.1793   *** 
## log(timepoint_mean)   0.0992    ** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
87 (85.3 to 88.8) 90.7 (88.4 to 93.1) 95.8 (90.2 to 101.7) 51 months 55 months
## [1] 34.01645

6. 6m timed Hop

Specify covariance matrix

Model Selection

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.     AICc.
## 1     Linear 45.28073 -90.56147 -82.56147 -76.22739 -81.27115
## 2        Log 49.05946 -98.11892 -90.11892 -83.78484 -88.82860
## 3   Poly (2) 45.91722 -91.83444 -81.83444 -74.05770 -79.76547
## 4 3 knot RCS 47.91608 -95.83217 -85.83217 -78.05543 -83.76320
## 5 4 knot RCS 47.00637 -94.01275 -82.01275 -72.85458 -78.90164
## 
## [[2]]

Final Model - 6m timed hop

## 
## Multivariate Meta-Analysis Model (k = 38; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 25)
## inner factor: timepoint_mean (nlvls = 29)
## 
##             estim    sqrt  fixed 
## tau^2      0.0034  0.0584     no 
## rho        0.0529             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 36) = 810.6433, p-val < .0001
## 
## Number of estimates:   38
## Number of clusters:    25
## Estimates per cluster: 1-4 (mean: 1.52, median: 1)
## 
## Test of Moderators (coefficient 2):¹
## F(df1 = 1, df2 = 7.45) = 19.0457, p-val = 0.0028
## 
## Model Results:
## 
##                      estimate      se¹     tval¹    df¹    pval¹    ci.lb¹ 
## intrcpt               -0.2170  0.0365   -5.9410   9.41   0.0002   -0.2991  
## log(timepoint_mean)    0.0566  0.0130    4.3641   7.45   0.0028    0.0263  
##                        ci.ub¹      
## intrcpt              -0.1349   *** 
## log(timepoint_mean)   0.0870    ** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
92.7 (90.6 to 94.8) 96.4 (93.7 to 99.1) 101.5 (96.5 to 106.8) 27 months 94 months
## [1] 36.02217

Case Control

Significantly less data available for between person/case control comparisons. Only single hop has enough data to run same analysis as within person.

1. Single Hop - Between Person

Specify covariance matrix

Model Selection

## [[1]]
##          mod  logLik. deviance.      AIC.      BIC.      AICc.
## 1     Linear 15.42067 -30.84133 -22.84133 -19.75098 -19.204971
## 2        Log 16.82728 -33.65455 -25.65455 -22.56420 -22.018191
## 3   Poly (2) 15.23835 -30.47671 -20.47671 -16.93645 -13.810039
## 4 3 knot RCS 17.66685 -35.33370 -25.33370 -21.79345 -18.667032
## 5 4 knot RCS 15.82739 -31.65477 -19.65477 -15.82043  -7.654771
## 
## [[2]]

Final Model - Single Leg Hop

## 
## Multivariate Meta-Analysis Model (k = 18; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort         (nlvls = 15)
## inner factor: timepoint_mean (nlvls = 17)
## 
##             estim    sqrt  fixed 
## tau^2      0.0070  0.0835     no 
## rho        1.0000             no 
## 
## Test for Residual Heterogeneity:
## QE(df = 16) = 127.1249, p-val < .0001
## 
## Number of estimates:   18
## Number of clusters:    15
## Estimates per cluster: 1-3 (mean: 1.20, median: 1)
## 
## Test of Moderators (coefficient 2):¹
## F(df1 = 1, df2 = 2.67) = 5.5065, p-val = 0.1114
## 
## Model Results:
## 
##                      estimate      se¹     tval¹    df¹    pval¹    ci.lb¹ 
## intrcpt               -0.2924  0.0722   -4.0512   3.59   0.0192   -0.5021  
## log(timepoint_mean)    0.0606  0.0258    2.3466   2.67   0.1114   -0.0276  
##                        ci.ub¹    
## intrcpt              -0.0827   * 
## log(timepoint_mean)   0.1488     
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 1) results based on cluster-robust inference (var-cov estimator: CR2,
##    approx t/F-tests and confidence intervals, df: Satterthwaite approx)
## Profiling tau2 = 1

## Profiling rho = 1

1 year 2 years 5 years Zero crossing Last Data Point
86.8 (82.4 to 91.4) 90.5 (85.1 to 96.3) 95.7 (84.9 to 107.8) 34 months 62 months
## [1] 7.725032

2. Triple Hop / Triple Crossover Hop - Between Person

Not enough data to be able to run longitudinal analysis. Reverting to univariate meta analysis.

##                  ROM           95%-CI %W(common) %W(random)
## Casp 2021     0.8750 [0.8287; 0.9239]       33.5       33.5
## Gokeler 2017a 0.8784 [0.8353; 0.9237]       39.0       39.0
## Kline 2018    0.8035 [0.7088; 0.9108]        6.3        6.3
## Norte 2020    0.8985 [0.8392; 0.9619]       21.2       21.2
## 
## Number of studies: k = 4
## 
##                         ROM           95%-CI     z  p-value
## Common effect model  0.8765 [0.8494; 0.9045] -8.22 < 0.0001
## Random effects model 0.8765 [0.8494; 0.9045] -8.21 < 0.0001
## 
## Quantifying heterogeneity:
##  tau^2 < 0.0001 [0.0000; 0.0303]; tau = 0.0007 [0.0000; 0.1741]
##  I^2 = 0.0% [0.0%; 84.7%]; H = 1.00 [1.00; 2.56]
## 
## Test of heterogeneity:
##     Q d.f. p-value
##  2.37    3  0.4995
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2
## - Q-Profile method for confidence interval of tau^2 and tau

##                    ROM           95%-CI %W(common) %W(random)
## Patterson 2020a 0.8349 [0.7469; 0.9333]       27.3       27.5
## Kline 2018      0.8161 [0.7011; 0.9501]       14.7       14.9
## Norte 2020      0.8934 [0.8276; 0.9645]       58.0       57.6
## 
## Number of studies: k = 3
## 
##                         ROM           95%-CI     z  p-value
## Common effect model  0.8654 [0.8165; 0.9174] -4.86 < 0.0001
## Random effects model 0.8652 [0.8157; 0.9176] -4.82 < 0.0001
## 
## Quantifying heterogeneity:
##  tau^2 < 0.0001 [0.0000; 0.0845]; tau = 0.0064 [0.0000; 0.2907]
##  I^2 = 0.0% [0.0%; 89.6%]; H = 1.00 [1.00; 3.10]
## 
## Test of heterogeneity:
##     Q d.f. p-value
##  1.64    2  0.4412
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2
## - Q-Profile method for confidence interval of tau^2 and tau

3. Side Hop - Between Person

Not enough data to be able to run longitudinal analysis. Reverting to univariate meta analysis.

  • Only 4 studies, 1 with multiple timepoint (Patterson)
  • 3 <24m, 2 > 50m.
##                           ROM           95%-CI %W(common) %W(random)
## Patterson 2020a        0.7080 [0.5807; 0.8632]       12.3       18.4
## Engelen-VanMelick 2017 0.8067 [0.6709; 0.9700]       14.3       20.1
## Falstrom 2017          0.9282 [0.8161; 1.0557]       29.3       28.5
## Faltstrom 2021         0.9032 [0.8133; 1.0030]       44.1       33.0
## 
## Number of studies: k = 4
## 
##                         ROM           95%-CI     z  p-value
## Common effect model  0.8694 [0.8109; 0.9321] -3.94 < 0.0001
## Random effects model 0.8508 [0.7631; 0.9486] -2.91   0.0036
## 
## Quantifying heterogeneity:
##  tau^2 = 0.0065 [0.0000; 0.2044]; tau = 0.0805 [0.0000; 0.4521]
##  I^2 = 52.1% [0.0%; 84.2%]; H = 1.44 [1.00; 2.51]
## 
## Test of heterogeneity:
##     Q d.f. p-value
##  6.26    3  0.0997
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2
## - Q-Profile method for confidence interval of tau^2 and tau

4. Vertical Hop - Between Person

Not enough data to be able to run longitudinal analysis. Reverting to univariate meta analysis.

  • Only 5 studies, varied timepoints
##                           ROM           95%-CI %W(common) %W(random)
## Engelen-VanMelick 2017 0.9492 [0.8467; 1.0641]       17.0       23.5
## Laudner 2015           0.8718 [0.7796; 0.9748]       17.8       23.8
## Markstrom 2023         1.0000 [0.9263; 1.0796]       37.8       26.9
## O'Malley 2018          0.7706 [0.7042; 0.8432]       27.4       25.8
## 
## Number of studies: k = 4
## 
##                         ROM           95%-CI     z  p-value
## Common effect model  0.9007 [0.8592; 0.9442] -4.35 < 0.0001
## Random effects model 0.8940 [0.7963; 1.0037] -1.90   0.0577
## 
## Quantifying heterogeneity:
##  tau^2 = 0.0114 [0.0022; 0.1779]; tau = 0.1068 [0.0466; 0.4218]
##  I^2 = 84.9% [62.3%; 93.9%]; H = 2.57 [1.63; 4.05]
## 
## Test of heterogeneity:
##      Q d.f. p-value
##  19.82    3  0.0002
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2
## - Q-Profile method for confidence interval of tau^2 and tau

5. 6m timed Hop - Between Person

Not enough data to be able to run longitudinal analysis. Reverting to univariate meta analysis.

  • Only 2 studies
##               ROM           95%-CI %W(common) %W(random)
## Kline 2018 0.6970 [0.5287; 0.9188]       13.0       30.3
## Norte 2020 0.8571 [0.7701; 0.9540]       87.0       69.7
## 
## Number of studies: k = 2
## 
##                         ROM           95%-CI     z p-value
## Common effect model  0.8343 [0.7551; 0.9219] -3.56  0.0004
## Random effects model 0.8051 [0.6683; 0.9700] -2.28  0.0226
## 
## Quantifying heterogeneity:
##  tau^2 = 0.0100; tau = 0.0998; I^2 = 46.6%; H = 1.37
## 
## Test of heterogeneity:
##     Q d.f. p-value
##  1.87    1  0.1713
## 
## Details on meta-analytical method:
## - Inverse variance method
## - Restricted maximum-likelihood estimator for tau^2

Regression of True Effects

Using studies which measure 2 hop tests concurrently, we run a bivariate analysis.

We have to account for non-independence of samples in the analysis conducting a variance/covariance matrix and also fitting random effects for this (type of outcome: within/between person nested within cohorts). To calculate this we assumed a rho for each hop test.

Single hop and other forward hops - 0.7 Single hop and side hop/vertical hop - 0.7

Sensitivity analyses show that estimates are stable to different values for this.

Single hop and Triple hop

## 
## Multivariate Meta-Analysis Model (k = 62; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 31)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed       level 
## tau^2.1    0.0014  0.0374     31     no  triple hop 
## tau^2.2    0.0016  0.0404     31     no  single hop 
## 
##             rho.trph  rho.sngh    trph  sngh 
## triple hop         1                 -    31 
## single hop    0.9608         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 60) = 797.9911, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 119.3674, p-val < .0001
## 
## Model Results:
## 
##                    estimate      se      zval    pval    ci.lb    ci.ub      
## measuretriple hop   -0.0732  0.0070  -10.3966  <.0001  -0.0870  -0.0594  *** 
## measuresingle hop   -0.0833  0.0076  -10.9208  <.0001  -0.0982  -0.0683  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##             estimate      se     zval    pval    ci.lb   ci.ub      
## intrcpt      -0.0469  0.0713  -0.6576  0.5108  -0.1865  0.0928      
## triple hop    1.0404  0.0767  13.5705  <.0001   0.8901  1.1906  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]

Single hop and Vertical hop

## 
## Multivariate Meta-Analysis Model (k = 22; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 10)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed         level 
## tau^2.1    0.0014  0.0381     11     no  vertical hop 
## tau^2.2    0.0004  0.0203     11     no    single hop 
## 
##               rho.vrth  rho.sngh    vrth  sngh 
## vertical hop         1                 -    10 
## single hop      0.2696         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 20) = 187.8013, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 118.6979, p-val < .0001
## 
## Model Results:
## 
##                      estimate      se     zval    pval    ci.lb    ci.ub      
## measurevertical hop   -0.1172  0.0130  -9.0397  <.0001  -0.1426  -0.0918  *** 
## measuresingle hop     -0.0626  0.0072  -8.6427  <.0001  -0.0768  -0.0484  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##               estimate      se    zval    pval    ci.lb   ci.ub      
## intrcpt         0.8118  0.1897  4.2798  <.0001   0.4400  1.1836  *** 
## vertical hop    0.1434  0.2133  0.6724  0.5013  -0.2746  0.5614      
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]

Single hop and Side hop

## 
## Multivariate Meta-Analysis Model (k = 26; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 12)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed       level 
## tau^2.1    0.0033  0.0570     13     no    side hop 
## tau^2.2    0.0015  0.0386     13     no  single hop 
## 
##             rho.sdhp  rho.sngh    sdhp  sngh 
## side hop           1                 -    12 
## single hop    0.8195         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 24) = 367.6610, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 27.3344, p-val < .0001
## 
## Model Results:
## 
##                    estimate      se     zval    pval    ci.lb    ci.ub      
## measureside hop     -0.0893  0.0175  -5.1129  <.0001  -0.1235  -0.0551  *** 
## measuresingle hop   -0.0553  0.0117  -4.7347  <.0001  -0.0782  -0.0324  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##           estimate      se    zval    pval   ci.lb   ci.ub      
## intrcpt     0.4395  0.1282  3.4281  0.0006  0.1882  0.6908  *** 
## side hop    0.5540  0.1402  3.9518  <.0001  0.2792  0.8288  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]

Single hop and 6 timed hop

## 
## Multivariate Meta-Analysis Model (k = 38; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 19)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed         level 
## tau^2.1    0.0020  0.0442     19     no  6m timed hop 
## tau^2.2    0.0013  0.0364     19     no    single hop 
## 
##               rho.6mth  rho.sngh    6mth  sngh 
## 6m timed hop         1                 -    19 
## single hop      0.8733         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 36) = 438.3183, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 86.5826, p-val < .0001
## 
## Model Results:
## 
##                      estimate      se     zval    pval    ci.lb    ci.ub      
## measure6m timed hop   -0.0732  0.0108  -6.7845  <.0001  -0.0944  -0.0521  *** 
## measuresingle hop     -0.0811  0.0089  -9.1511  <.0001  -0.0985  -0.0637  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##               estimate      se    zval    pval   ci.lb   ci.ub      
## intrcpt         0.2548  0.1252  2.0347  0.0419  0.0094  0.5002    * 
## 6m timed hop    0.7180  0.1347  5.3289  <.0001  0.4539  0.9821  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]

Triple hop and Triple crossover hop

## 
## Multivariate Meta-Analysis Model (k = 40; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 20)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed                 level 
## tau^2.1    0.0020  0.0445     20     no  triple crossover hop 
## tau^2.2    0.0019  0.0441     20     no            triple hop 
## 
##                       rho.trch  rho.trph    trch  trph 
## triple crossover hop         1                 -    20 
## triple hop              0.9819         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 38) = 842.6705, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 61.3503, p-val < .0001
## 
## Model Results:
## 
##                              estimate      se     zval    pval    ci.lb 
## measuretriple crossover hop   -0.0779  0.0104  -7.4925  <.0001  -0.0983 
## measuretriple hop             -0.0802  0.0103  -7.8198  <.0001  -0.1003 
##                                ci.ub      
## measuretriple crossover hop  -0.0575  *** 
## measuretriple hop            -0.0601  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##                       estimate      se     zval    pval    ci.lb   ci.ub      
## intrcpt                 0.0217  0.0564   0.3858  0.6996  -0.0887  0.1322      
## triple crossover hop    0.9742  0.0609  15.9881  <.0001   0.8548  1.0937  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]

Triple crossover and 6m

## 
## Multivariate Meta-Analysis Model (k = 32; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 16)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed                 level 
## tau^2.1    0.0027  0.0516     16     no          6m timed hop 
## tau^2.2    0.0016  0.0403     16     no  triple crossover hop 
## 
##                       rho.6mth  rho.trch    6mth  trch 
## 6m timed hop                 1                 -    16 
## triple crossover hop    0.7293         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 30) = 727.9863, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 49.2613, p-val < .0001
## 
## Model Results:
## 
##                              estimate      se     zval    pval    ci.lb 
## measure6m timed hop           -0.0754  0.0137  -5.5179  <.0001  -0.1022 
## measuretriple crossover hop   -0.0747  0.0107  -6.9924  <.0001  -0.0956 
##                                ci.ub      
## measure6m timed hop          -0.0486  *** 
## measuretriple crossover hop  -0.0537  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##               estimate      se    zval    pval   ci.lb   ci.ub      
## intrcpt         0.3999  0.1453  2.7526  0.0059  0.1152  0.6847   ** 
## 6m timed hop    0.5695  0.1567  3.6350  0.0003  0.2624  0.8766  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]

Triple hop and 6m

## 
## Multivariate Meta-Analysis Model (k = 32; method: REML)
## 
## Variance Components:
## 
## outer factor: cohort  (nlvls = 16)
## inner factor: measure (nlvls = 2)
## 
##             estim    sqrt  k.lvl  fixed         level 
## tau^2.1    0.0026  0.0505     16     no  6m timed hop 
## tau^2.2    0.0013  0.0365     16     no    triple hop 
## 
##               rho.6mth  rho.trph    6mth  trph 
## 6m timed hop         1                 -    16 
## triple hop      0.8079         1      no     - 
## 
## Test for Residual Heterogeneity:
## QE(df = 30) = 548.8912, p-val < .0001
## 
## Test of Moderators (coefficients 1:2):
## QM(df = 2) = 58.4267, p-val < .0001
## 
## Model Results:
## 
##                      estimate      se     zval    pval    ci.lb    ci.ub      
## measure6m timed hop   -0.0763  0.0133  -5.7132  <.0001  -0.1024  -0.0501  *** 
## measuretriple hop     -0.0728  0.0096  -7.6154  <.0001  -0.0916  -0.0541  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## [[1]]
## 
##               estimate      se    zval    pval   ci.lb   ci.ub      
## intrcpt         0.3891  0.1207  3.2244  0.0013  0.1526  0.6257   ** 
## 6m timed hop    0.5835  0.1302  4.4798  <.0001  0.3282  0.8387  *** 
## 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## 
## [[2]]